Single-Phase AC Power Circuits - LabVolt Series by Festo Didactic AC Power... · Single-Phase AC Power Circuits Course Sample. 594089 . Order no.: 594089 (Printed version) 594404

Electricity and New Energy

Single-Phase AC Power Circuits

Course Sample 594089

Order no.: 594089 (Printed version) 594404 (CD-ROM) First Edition Revision level: 06/2018

By the staff of Festo Didactic

© Festo Didactic Ltée/Ltd, Quebec, Canada 2018 Internet: www.festo-didactic.com e-mail: [email protected]

Printed in Canada All rights reserved ISBN 978-2-89747-806-3 (Printed version) ISBN 978-2-89747-807-0 (CD-ROM) Legal Deposit – Bibliothèque et Archives nationales du Québec, 2018 Legal Deposit – Library and Archives Canada, 2018

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The purchaser shall be entitled to use the work to train his/her staff at the purchaser’s site/location and shall also be entitled to use parts of the copyright material as the basis for the production of his/her own training documentation for the training of his/her staff at the purchaser’s site/location with acknowledgement of source and to make copies for this purpose. In the case of schools/technical colleges, training centers, and universities, the right of use shall also include use by school and college students and trainees at the purchaser’s site/location for teaching purposes.

The right of use shall in all cases exclude the right to publish the copyright material or to make this available for use on intranet, Internet, and LMS platforms and databases such as Moodle, which allow access by a wide variety of users, including those outside of the purchaser’s site/location.

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© Festo Didactic 594089 III

Safety and Common Symbols

The following safety and common symbols may be used in this course and on the equipment:

Symbol Description

DANGER indicates a hazard with a high level of risk which, if not avoided, will result in death or serious injury.

WARNING indicates a hazard with a medium level of risk which, if not avoided, could result in death or serious injury.

CAUTION indicates a hazard with a low level of risk which, if not avoided, could result in minor or moderate injury.

CAUTION used without the Caution, risk of danger sign , indicates a hazard with a potentially hazardous situation which, if not avoided, may result in property damage.

Caution, risk of electric shock

Caution, hot surface

Caution, risk of danger. Consult the relevant user documentation.

Caution, lifting hazard

Caution, belt drive entanglement hazard

Caution, chain drive entanglement hazard

Caution, gear entanglement hazard

Caution, hand crushing hazard

Notice, non-ionizing radiation

Consult the relevant user documentation.

Direct current

Alternating current

Safety and Common Symbols

IV © Festo Didactic 594089

Symbol Description

Both direct and alternating current

Three-phase alternating current

Earth (ground) terminal

Protective conductor terminal

Frame or chassis terminal

Equipotentiality

On (supply)

Off (supply)

Equipment protected throughout by double insulation or reinforced insulation

In position of a bi-stable push control

Out position of a bi-stable push control

© Festo Didactic 594089 V

Table of Contents

Preface .................................................................................................................. XI

About This Course ..............................................................................................XIII

To the Instructor .................................................................................................. XV

Unit 1 Alternating Current ........................................................................ 1

DISCUSSION OF FUNDAMENTALS ......................................................... 1 DC circuits versus ac circuits .................................................... 1 Alternating current (ac) and ac voltage ..................................... 2 Alternating current and ac voltage supplied by public power distribution utilities .......................................................... 2 Safety rules ............................................................................... 3

Ex. 1-1 The Sine Wave ............................................................................... 5

DISCUSSION ...................................................................................... 5 Relationship between a rotating phasor and a sine wave ........ 5 Period and frequency of a sinusoidal voltage or current ........... 8 Amplitude and instantaneous value of a sinusoidal voltage or current ................................................................................... 8 Effective or root-mean-square (rms) value and heating capacity ................................................................................... 10 Effective (rms) value of a sinusoidal voltage or current .......... 10

PROCEDURE .................................................................................... 11 Setup and connections ............................................................ 11 Measuring voltage, current, and frequency in an ac circuit .... 13 Relationship between frequency and period ........................... 15 Measuring voltage, current, and frequency in a series ac circuit .................................................................................. 16

CONCLUSION ................................................................................... 19

REVIEW QUESTIONS ......................................................................... 19

Ex. 1-2 Phase Angle and Phase Shift ..................................................... 21

DISCUSSION .................................................................................... 21 Phase angle ............................................................................ 21 Phase shift ............................................................................... 23

PROCEDURE .................................................................................... 25 Setup and connections ............................................................ 25 Measuring the phase shift between two voltage sine waves in a resistor-inductor (RL) circuit .................................. 26 Measuring the phase shift between two voltage sine waves in a resistor-capacitor (RC) circuit ............................... 30

CONCLUSION ................................................................................... 34


Table of Contents

VI © Festo Didactic 594089

Ex. 1-3 Instantaneous Power and Average Power ................................ 37

DISCUSSION .................................................................................... 37 Instantaneous power ............................................................... 37 Average power ........................................................................ 38 Rationale behind rms values ................................................... 39

PROCEDURE .................................................................................... 40 Setup and connections ............................................................ 40 Average power measurements ............................................... 41 Rationale behind rms values ................................................... 44

CONCLUSION ................................................................................... 46


Unit 2 Resistance, Reactance, and Impedance ................................... 49

DISCUSSION OF FUNDAMENTALS ....................................................... 49 Introduction to inductors and capacitors ................................. 49 Distinction between resistance, reactance, and impedance ............................................................................... 50

Ex. 2-1 Inductive Reactance .................................................................... 51

DISCUSSION .................................................................................... 51 Inductors and inductive reactance .......................................... 51 Inductive phase shift................................................................ 52

PROCEDURE .................................................................................... 52 Setup and connections ............................................................ 52 Inductance and inductive reactance ........................................ 54 Effect of frequency on the inductive reactance ....................... 56 Measuring the inductive phase shift ........................................ 57

CONCLUSION ................................................................................... 59


Ex. 2-2 Capacitive Reactance .................................................................. 61

DISCUSSION .................................................................................... 61 Capacitors and capacitive reactance ...................................... 61 Capacitive phase shift ............................................................. 62

PROCEDURE .................................................................................... 62 Setup and connections ............................................................ 62 Capacitance and capacitive reactance ................................... 64 Effect of the frequency on the capacitive reactance ............... 66 Measuring the capacitive phase shift ...................................... 67

CONCLUSION ................................................................................... 69


Table of Contents

© Festo Didactic 594089 VII

Ex. 2-3 Impedance .................................................................................... 71

DISCUSSION .................................................................................... 71 Phasor diagrams related to resistors, inductors, and capacitors ................................................................................ 71

Phasor diagram related to a resistor .......................................... 72 Phasor diagram related to an inductor ....................................... 72 Phasor diagram related to a capacitor ....................................... 73

Equivalent reactance of series-connected reactive components ............................................................................. 73 Impedance of resistors, inductors, and capacitors connected in series ................................................................. 75 Impedance of resistors, inductors, and capacitors connected in parallel ............................................................... 78

PROCEDURE .................................................................................... 79 Setup and connections ............................................................ 79 Equivalent reactance of a series LC circuit ............................. 80 Impedance of a series RL circuit ............................................. 83 Impedance of a series RC circuit ............................................ 84 Impedance of a series RLC circuit .......................................... 86 Impedance of a parallel RL circuit ........................................... 87 Impedance of a parallel RC circuit .......................................... 88

CONCLUSION ................................................................................... 90


Unit 3 Power in AC Circuits ................................................................... 95

DISCUSSION OF FUNDAMENTALS ....................................................... 95 Introduction to active, reactive, and apparent power .............. 95

Ex. 3-1 Active and Reactive Power ......................................................... 97

DISCUSSION .................................................................................... 97 Active power in a resistor ........................................................ 97 Reactive power in an inductor ................................................. 98 Reactive power in a capacitor ................................................. 99 Power Meter .......................................................................... 100

PROCEDURE .................................................................................. 100 Setup and connections .......................................................... 101 Active power in a resistor ...................................................... 102 Reactive power in an inductor ............................................... 104 Reactive power in a capacitor ............................................... 107

CONCLUSION ................................................................................. 109

REVIEW QUESTIONS ....................................................................... 109

Table of Contents

VIII © Festo Didactic 594089

Ex. 3-2 Apparent Power and the Power Triangle ................................ 111

DISCUSSION .................................................................................. 111 Phasor diagrams related to active and reactive power ......... 111

Phasor diagram related to the active power in a resistor ......... 111 Phasor diagram related to the reactive power in an inductor... 112 Phasor diagram related to the reactive power in a capacitor ... 112

Apparent power ..................................................................... 113 Power triangle ....................................................................... 114 Power factor .......................................................................... 114

PROCEDURE .................................................................................. 115 Setup and connections .......................................................... 115 Total reactive power in a circuit ............................................. 116 Apparent power, power factor, and power triangle ............... 119

CONCLUSION ................................................................................. 123


Unit 4 AC Circuits Analysis ................................................................. 129

DISCUSSION OF FUNDAMENTALS ..................................................... 129 Introduction to circuit analysis ............................................... 129 Circuit analysis methods ....................................................... 129

Ex. 4-1 Solving Simple AC Circuits Using Circuit Impedance Calculation .................................................................................. 131

DISCUSSION .................................................................................. 131 Solving simple parallel circuits .............................................. 131

Example .................................................................................. 132 Solving simple series circuits ................................................ 133

Example .................................................................................. 134

PROCEDURE .................................................................................. 135 Setup and connections .......................................................... 135 Solving a simple parallel ac circuit ........................................ 136 Solving a simple series ac circuit .......................................... 138

CONCLUSION ................................................................................. 141


Ex. 4-2 Solving AC Circuits Using the Power Triangle Method ......... 143

DISCUSSION .................................................................................. 143 Solving ac circuits using the power triangle method ............. 143

Example .................................................................................. 144

PROCEDURE .................................................................................. 146 Setup and connections .......................................................... 147 Solving an ac circuit using the power triangle method .......... 148

CONCLUSION ................................................................................. 152


Table of Contents

© Festo Didactic 594089 IX

Appendix A Equipment Utilization Chart...................................................... 159

Appendix B Glossary of New Terms ............................................................. 161

Appendix C Impedance Table for the Load Modules .................................. 165

Appendix D Vectorial Calculations ............................................................... 167 Vectorial division ................................................................... 168 Vectorial multiplication .......................................................... 169

Appendix E Circuit Diagram Symbols .......................................................... 171

Appendix F Course Completion with the AC 230V/DC 325V Variable Power Supply ............................................................................. 177

Index of New Terms ........................................................................................... 179

Bibliography ........................................................................................................ 181

© Festo Didactic 594089 XI

Preface

Electrical energy is part of our life since more than a century and the number of applications using electric power keeps increasing. This phenomenon is illustrated by the steady growth in electric power demand observed worldwide. In reaction to this phenomenon, the production of electrical energy using renewable natural resources (e.g., wind, sunlight, rain, tides, geothermal heat, etc.) has gained much importance in recent years since it helps to meet the increasing demand for electric power and is an effective means of reducing greenhouse gas (GHG) emissions.

To help answer the increasing needs for training in the wide field of electrical energy, Festo Didactic developed a series of modular courses. These courses are shown below as a flow chart, with each box in the flow chart representing a course.

Festo Didactic courses in electrical energy.

Teaching includes a series of courses providing in-depth coverage of basic topics related to the field of electrical energy such as dc power circuits, ac power circuits, and power transformers. Other courses also provide in-depth coverage of solar power and wind power. Finally, two courses deal with photovoltaic systems and wind power systems, with focus on practical aspects related to these systems.

We invite readers of this manual to send us their tips, feedback, and suggestions for improving the book.

Please send these to [email protected].

The authors and Festo Didactic look forward to your comments.

mailto:[email protected]

© Festo Didactic 594089 XIII

About This Course

Alternating-current (ac) power systems began to develop quickly in the late 19th century, following key developments in the field of electricity, mainly the invention of the polyphase system of electrical distribution by scientist Nikola Tesla, and the development of mathematical analysis of electricity by Charles Steinmetz, James Clerk Maxwell, and William Thomson (Lord Kelvin).

The main advantage of ac power systems is that high amounts of power can be transmitted efficiently over long transmission lines. Step-up transformers are used at the ac power generating point to increase the voltage and decrease the current. The power lost as heat in the resistance of a transmission line increases by the square of the current. Therefore, ac power is transmitted at very high voltages and low currents to reduce power losses in the line resistance to a minimum. At the receiving end of the line, step-down transformers reduce the voltage and increase the current to levels compatible with residential or industrial equipment.

Today ac power systems are used throughout the world for driving motors and powering electric equipment in transport, heating, lighting, communications, and computation.

This course, Single-Phase AC Power Circuit, introduces students to the fundamentals of alternating current, such as the sine wave, period and frequency, phase angle and phase shift, instantaneous and average power, etc. Students then become familiar with the inductor and capacitor. The course continues with more advanced topics such as the impedance, active power, reactive power, apparent power, and power triangle. The course concludes by teaching students how to solve ac power circuits using the impedance calculation method or the power triangle method.

Most lighting in urban centers is powered using single-phase alternative current.

About This Course

XIV © Festo Didactic 594089

Completion of the hands-on exercises in the present course requires the 4 Quadrant Power Supply and Dynamometer Controller. In these exercises, the 4 Quadrant Power Supply and Dynamometer Controller is mostly used as a variable voltage and frequency, ac power source. The 4 Quadrant Power Supply and Dynamometer Controller can be replaced with the AC 230V/DC 325V Variable Power Supply (variable-voltage, ac/dc power source) and two multimeters to perform the majority of the manipulations in the hands-on exercises of the present course. Appendix F shows how to use the AC 230V/DC 325V Variable Power Supply and the two multimeters in place of the 4 Quadrant Power Supply and Dynamometer Controller to perform the hands-on exercises.

Safety considerations

Safety symbols that may be used in this course and on the equipment are listed in the Safety and Common Symbols table at the beginning of this document.

Safety procedures related to the tasks that you will be asked to perform are indicated in each exercise.

Make sure that you are wearing appropriate protective equipment when performing the tasks. You should never perform a task if you have any reason to think that a manipulation could be dangerous for you or your teammates.

Prerequisite

As a prerequisite to this course, you should have completed course DC Power Circuits.

Systems of units

Units are expressed using the International System of Units (SI).

© Festo Didactic 594089 XV

To the Instructor

You will find in this Instructor version of the course all the elements included in the Student version of the course together with the answers to all questions, results of measurements, graphs, explanations, suggestions, and, in some cases, instructions to help you guide the students through their learning process. All the information that applies to you is placed between markers and appears in red.

Accuracy of measurements

The numerical results of the hands-on exercises may differ from one student to another. For this reason, the results and answers given in this course should be considered as a guide. Students who correctly performed the exercises should expect to demonstrate the principles involved and make observations and measurements similar to those given as answers.

Equipment installation and use

In order for students to be able to safely perform the hands-on exercises in this course, the equipment must have been properly installed, i.e., according to the instructions given in the accompanying Safety Instructions and Commissioning manual. Also, the students must familiarize themselves with the safety directives provided in the Safety Instructions and Commissioning manual and observe these directives when using the equipment.

Sample

Extracted from

Instructor Guide

© Festo Didactic 594089 21

When you have completed this exercise, you will know what a phase angle is and how the phase angle modifies the initial displacement of a sine wave. You will be able to determine the phase shift between two sine waves, either by comparing their phase angles or by determining their separation in time. You will also know how to distinguish a leading phase shift from a lagging phase shift.

The Discussion of this exercise covers the following points:

Phase angle Phase shift

Phase angle

As you have seen in Exercise 1-1, the graphical representation of a sine wave can be expressed by the following equation:

𝑎𝑎(𝑡𝑡) = 𝐴𝐴 sin(𝜔𝜔𝑡𝑡) (1-8)

where 𝑎𝑎(𝑡𝑡) is the instantaneous value of the sine wave at a given instant 𝑡𝑡. 𝐴𝐴 is the amplitude of the sine wave. 𝜔𝜔 is the angular velocity, expressed in radians per second (rad/s). 𝑡𝑡 is the time, expressed in seconds (s).

This equation assumes that the sine wave cycle begins at the exact moment when 𝑡𝑡 = 0 (as is shown in Figure 1-10). As you will see later, this is not always the case. To represent the initial position of the sine wave, the notion of phase angle 𝜃𝜃 is introduced in the equation below:

𝑎𝑎(𝑡𝑡) = 𝐴𝐴 sin(𝜔𝜔𝑡𝑡 + 𝜃𝜃) (1-9)

where 𝜃𝜃 is the phase angle of the sine wave, expressed in degrees (°) or

radians (rad).

From Equation (1-9), it is easy to observe that the initial value (i.e., the value at 𝑡𝑡 = 0) of the sine wave depends entirely on the phase angle 𝜃𝜃 because the term 𝜔𝜔𝑡𝑡 equals 0 at 𝑡𝑡 = 0. In other words, the phase angle 𝜃𝜃 determines by how much the value of a sine wave differs from 0 at time 𝑡𝑡 = 0, and thus, the position in time of the sine wave.

Phase Angle and Phase Shift

Exercise 1-2

EXERCISE OBJECTIVE

DISCUSSION OUTLINE

DISCUSSION

Ex. 1-2 – Phase Angle and Phase Shift Discussion

22 © Festo Didactic 594089

Figure 1-10 shows a sine wave with a phase angle 𝜃𝜃 of 0°. The initial value of this sine wave is 0 because 𝐴𝐴 sin(𝜔𝜔 ∙ 0 + 0) = 0. This sine wave is identical to those seen in Exercise 1-1, as a phase angle value of 0° was implied by the absence of 𝜃𝜃 in the equations given in Exercise 1-1.

Figure 1-10. Sine wave with a phase angle 𝜽𝜽 of 0°.

Figure 1-11 shows a sine wave with a phase angle 𝜃𝜃 of 45°. As you can see from the figure, a positive phase angle (0° to 180°) results in the sine wave having a positive instantaneous value when 𝑡𝑡 = 0. In other words, a positive phase angle shifts the sine wave toward the left, i.e., advances the sine wave in time.

Figure 1-11. Sine wave with a phase angle 𝜽𝜽 of 45°.

Figure 1-12 shows a sine wave with a phase angle 𝜃𝜃 of -60°. A negative phase angle (0° to -180°) results in the sine wave having a negative instantaneous value when 𝑡𝑡 = 0. In other words, a negative phase angle shifts the sine wave toward the right, i.e., delays the sine wave in time.

Figure 1-12. Sine wave with a phase angle 𝜽𝜽 of -60°.

Figure 1-10 to Figure 1-12 also show the phasor representations of the sine waves at time 𝑡𝑡 = 0. Notice that, in each figure, the vertical distance between the tip of the rotating phasor representing the sine wave matches the instantaneous value of the sine wave at 𝑡𝑡 = 0.

𝐴𝐴

𝐴𝐴

𝐴𝐴

𝐴𝐴

𝐴𝐴

𝐴𝐴

𝑡𝑡

𝑡𝑡

𝑡𝑡

𝑡𝑡 = 0

𝑡𝑡 = 0

𝑡𝑡 = 0

𝜃𝜃 = 0°

𝜃𝜃 = 45°

𝜃𝜃 = -60°

Ex. 1-2 – Phase Angle and Phase Shift Discussion


Phase shift

When comparing two sine waves having the same frequency, the difference between their respective phase angles is called the phase shift and is expressed in degrees (°) or radians (rad). The magnitude of the phase shift indicates the extent of separation in time between the two sine waves, while the polarity of the phase shift (positive or negative) indicates the relationship in time between the two sine waves (leading or lagging).The sine wave amplitude value has no effect on the phase shift, as it does not change the period nor the frequency of the sine wave. Sine waves with different frequencies and, as an extension, different periods, cannot be compared by using their phase angles as their cycles do not correspond.

The phase shift between two sine waves is expressed as an angle representing a portion of a complete cycle of the sine waves. One of the two sine waves is used as the reference for phase shift measurements. The phase shift is calculated by subtracting the phase angle 𝜃𝜃𝑅𝑅𝑅𝑅𝑅𝑅. of the reference sine wave from the phase angle 𝜃𝜃 of the sine wave of interest. This is written as an equation below.

Phase shift = 𝜃𝜃 − 𝜃𝜃𝑅𝑅𝑅𝑅𝑅𝑅. (1-10)

where 𝜃𝜃 is the phase angle of the sine wave of interest, expressed in

degrees (°) or radians (rad). 𝜃𝜃𝑅𝑅𝑅𝑅𝑅𝑅. is the phase angle of the reference sine wave, expressed in

degrees (°) or radians (rad).

Figure 1-13 is an example showing how the phase shift between two sine waves (X and Y) can be calculated using their phase angles.

Figure 1-13. Phase shift between two sine waves with phase angles of 45° and -60°.

In the figure, sine wave X has a phase angle 𝜃𝜃 of 45°, while sine wave Y has a phase angle 𝜃𝜃 of -60°. Depending on which sine wave is used as the reference, the phase shift can be +105° or -105°. When sine wave X is considered as the reference, the phase shift of sine wave Y with respect to sine wave X is -105°(-60° - 45° = -105°). The minus sign in this phase shift value indicates that sine wave Y lags reference sine wave X. For this reason, this phase shift value can also be expressed as 105° lagging. Conversely, when sine wave Y is considered as the reference, the phase shift of sine wave X with respect to sine wave Y is +105° (45° - (-60)° = +105°). The plus sign in this phase shift value indicates that sine wave X leads reference sine wave Y. For this reason, this phase shift value can also be expressed as 105° leading. Note that whenever two sine waves have different phase angles, the phase shift value is not zero, and thus, these sine waves are said to be out of phase.

𝐴𝐴

𝐴𝐴

𝑡𝑡

𝑡𝑡 = 0

45° -60°

X Y

X

Y 105°

𝑡𝑡 = 0

Phase shift

Ex. 1-2 – Phase Angle and Phase Shift Procedure Outline


It is possible to determine the phase shift between two sine waves of the same frequency without knowing their respective phase angles 𝜃𝜃. The following equation is used:

Phase shift = 𝑑𝑑𝛵𝛵

× 360° = 𝑑𝑑𝛵𝛵

× 2𝜋𝜋 rad (1-11)

where 𝑑𝑑 is the time interval between a given reference point on each of the

two sine waves, expressed in seconds (s). 𝛵𝛵 is the period of the sine waves, expressed in seconds (s).

This equation shows in a concrete way why it is not possible to calculate the phase shift between two sine waves having different frequencies 𝑓𝑓, as a common period 𝛵𝛵 (𝛵𝛵 = 1 𝑓𝑓⁄ ) is needed for the equation to be valid.

Consider, for example, the sine waves shown in Figure 1-14. Using Equation (1-11), the phase shift between the two sine waves is equal to:

Phase shift =𝑑𝑑𝑇𝑇

× 360° =3.33 ms20.0 ms

× 360° = 60°

When sine wave 1 is used as the reference, the phase shift is lagging because sine wave 2 is delayed with respect to sine wave 1. Conversely, when sine wave 2 is considered as the reference, the phase shift is leading because sine wave 1 is in advance with respect to sine wave 2.

Figure 1-14. Phase shift between two sine waves having the same frequency.

The Procedure is divided into the following sections:

Setup and connections Measuring the phase shift between two voltage sine waves in a resistor-

inductor (RL) circuit Measuring the phase shift between two voltage sine waves in a

resistor-capacitor (RC) circuit

PROCEDURE OUTLINE

𝑇𝑇 20.0 ms

𝑑𝑑 3.33 ms

𝑡𝑡

Sine wave 1

Sine wave 2

Ex. 1-2 – Phase Angle and Phase Shift Procedure


High voltages are present in this laboratory exercise. Do not make or modify any banana jack connections with the power on unless otherwise specified.

Setup and connections

In this section, you will connect an ac circuit containing an inductor and a resistor in series and set up the equipment to measure the source voltage 𝑈𝑈𝑅𝑅, as well as the voltage across the resistor 𝑈𝑈𝑅𝑅.

1. Refer to the Equipment Utilization Chart in Appendix A to obtain the list of equipment required to perform this exercise.

a The 4 Quadrant Power Supply and Dynamometer Controller can be replaced with the AC 230V/DC 325V Variable Power Supply and two multimeters to perform the exercise. Appendix F shows how to use the AC 230V/DC 325V Variable Power Supply and the two multimeters in place of the 4 Quadrant Power Supply and Dynamometer Controller to perform the exercises.

2. Set up the equipment by performing the following tasks in the order they are listed.

Install the equipment required in the workstation.

Make the equipment earthing connections.

Connect the equipment to ac power outlets that are properly protected.

Connect the Data Acquisition and Control Interface to the AC 24V Power Supply.

Turn on (i.e., unlock) electric power at your workstation, if applicable.

Turn the equipment on.

Set the 4 Quadrant Power Supply and Dynamometer Controller for operation as a power supply.

Connect the equipment to USB ports of the host computer.

Turn the host computer on, then start the LVDAC-EMS software.

a Refer to the procedure of Exercise 1-1 for detailed manipulations related to the tasks above, if necessary.

3. In LVDAC-EMS, make sure that the Data Acquisition and Control Interface and the 4 Quadrant Power Supply and Dynamometer Controller are detected. Make sure that the Computer-Based Instrumentation function for the Data Acquisition and Control Interface is available. Select the network voltage and frequency that correspond to the voltage and frequency of your local ac power network.

PROCEDURE



4. Set up the circuit shown in Figure 1-15. This circuit contains a resistor 𝑅𝑅 and an inductor 𝐿𝐿. Inductors are studied in the next unit of this manual.

Figure 1-15. AC circuit with a resistor and an inductor.

The value of inductor 𝐿𝐿 in the circuit of Figure 1-15 is referred to as the inductance and is expressed in henries (H). The inductance value to be used is 1.00 H.

Make the necessary connections and switch settings on the Resistive Load in order to obtain the resistance value required.

a Appendix C of this manual lists the switch settings to implement on the Resistive Load in order to obtain various resistance values.

Make the necessary connections and switch settings on the Inductive Load in order to obtain the inductance value required.

a If necessary, ask your instructor to assist you to obtain the inductance value required.

Use inputs U1 and U2 of the Data Acquisition and Control Interface to measure the source voltage 𝑈𝑈𝑅𝑅 and the voltage across the resistor 𝑈𝑈𝑅𝑅, respectively.

5. In LVDAC-EMS, make the settings required to make the 4 Quadrant Power Supply and Dynamometer Controller operate as a variable voltage and frequency, ac power source. Then, set the ac power source as follows:

• Voltage: 100 V

• Frequency: same as the local ac power network frequency

At the moment, leave the ac power source off. The ac power source will be turned on in the next section of the procedure.

Measuring the phase shift between two voltage sine waves in a resistor-inductor (RL) circuit

In this section, you will observe the waveforms (sine waves) of the source voltage 𝑈𝑈𝑅𝑅 and the resistor voltage 𝑈𝑈𝑅𝑅 to determine the phase shift between the two sine wave voltages. Then, you will use a phasor analyzer to measure the phase shift between the source voltage phasor and the resistor voltage phasor, and compare it to the phase shift determined from the voltage waveforms.

a As you will see later, due to the presence of an inductor in the circuit, the circuit current lags behind the source voltage. As a result, the voltage 𝑈𝑈𝑅𝑅 measured across the resistor is out of phase with respect to the source voltage 𝑈𝑈𝑅𝑅.

𝑅𝑅 314 Ω

𝐿𝐿

𝑈𝑈𝑅𝑅 100 V U2 U1



6. In LVDAC-EMS, set meters U1 and U2 to measure the rms values of the source voltage 𝑈𝑈𝑅𝑅 and voltage 𝑈𝑈𝑅𝑅 across resistor 𝑅𝑅, respectively.

a When doing measurements using the instruments in LVDAC-EMS, always select the continuous refresh option. This ensures the instruments display updated data at all times.

In LVDAC-EMS, turn the ac power source on. Readjust the ac power source voltage 𝑈𝑈𝑅𝑅 (indicated by meter U1) so that it is equal to 100 V.

7. In LVDAC-EMS, display voltage 𝑈𝑈𝑅𝑅 (input U1) and voltage 𝑈𝑈𝑅𝑅 (input U2) on channels 1 and 2 of the oscilloscope, respectively. Set the time base so as to display at least two cycles of the waveforms. Place the traces of the two channels at the same vertical position.

8. In LVDAC-EMS, use the oscilloscope to measure the period 𝑇𝑇 of the source voltage 𝑈𝑈𝑅𝑅. Record the value below.

a To obtain an accurate measurement, use the vertical cursors of the LVDAC-EMS oscilloscope to measure the period or any other time interval.

Period 𝑇𝑇 = ms

Period 𝑇𝑇 = 20.05 ms. The results are shown in the following figure.

Period 𝑻𝑻 of the waveform of the source voltage 𝑼𝑼𝑺𝑺 at a frequency of 50 Hz.

Oscilloscope Settings Channel-1 Input .............................. U1 Channel-1 Scale ..................... 50 V/div Channel-1 Coupling ........................ DC Channel-2 Input .............................. U2 Channel-2 Scale ..................... 50 V/div Channel-2 Coupling ........................ DC Display Filtering .............................. On Show Cursors .......................... Vertical Trigger Type .......................... Software Time Base .............................. 5 ms/div Trigger Source .............................. Ch1 Trigger Level ...................................... 0 Trigger Slope ............................. Rising

Period 𝑇𝑇

Period 𝑇𝑇



9. In LVDAC-EMS, use the oscilloscope to measure the period 𝑇𝑇 of the resistor voltage 𝑈𝑈𝑅𝑅. Record the value below.



Period 𝑻𝑻 of the waveform of the resistor voltage 𝑼𝑼𝑹𝑹 at a frequency of 50 Hz.

10. Compare the period 𝑇𝑇 of the resistor voltage 𝑈𝑈𝑅𝑅 measured in the previous step with the period 𝑇𝑇 of the source voltage 𝑈𝑈𝑅𝑅 recorded in step 8. Are the values close to each other?

Yes No

Yes

11. In LVDAC-EMS, use the oscilloscope to measure the time interval 𝑑𝑑 between the waveforms of the source voltage 𝑈𝑈𝑅𝑅 and resistor voltage 𝑈𝑈𝑅𝑅. Record the value below.

Time interval 𝑑𝑑 = ms


Period 𝑇𝑇

Period 𝑇𝑇



Time interval 𝑑𝑑 = 2.25 ms. The results are shown in the following figure.

Time interval 𝒅𝒅 between the waveforms of the source voltage 𝑼𝑼𝑺𝑺 and resistor voltage 𝑼𝑼𝑹𝑹 at a frequency of 50 Hz.

12. Using Equation (1-11), calculate the phase shift between the source voltage 𝑈𝑈𝑅𝑅 and the resistor voltage 𝑈𝑈𝑅𝑅. Consider the source voltage waveform as the reference.

Phase shift = °

Phase shift = 𝑑𝑑𝛵𝛵∙ 360° = 2.25 ms

20.1 ms∙ 360° = 40.3°

Phase shift = 40.3° lagging or −40.3°

13. Is the resistor voltage 𝑈𝑈𝑅𝑅 leading or lagging the source voltage 𝑈𝑈𝑅𝑅?

The resistor voltage 𝑈𝑈𝑅𝑅 is lagging the source voltage 𝑈𝑈𝑅𝑅.

14. In LVDAC-EMS, use the phasor analyzer to display the phasors of source voltage 𝑈𝑈𝑅𝑅 (input U1) and resistor voltage 𝑈𝑈𝑅𝑅 (input U2). Select the source voltage 𝑈𝑈𝑅𝑅 (input U1) as the reference phasor. Measure the phase angles 𝜃𝜃𝑈𝑈𝑅𝑅 and 𝜃𝜃𝑈𝑈𝑅𝑅 of the voltage phasors.

Phase angle 𝜃𝜃𝑈𝑈𝑅𝑅 = °


Oscilloscope Settings Channel-1 Input .............................. U1 Channel-1 Scale ..................... 50 V/div Channel-1 Coupling ........................ DC Channel-2 Input .............................. U2 Channel-2 Scale ..................... 50 V/div Channel-2 Coupling ........................ DC Display Filtering .............................. On Show Cursors .......................... Vertical Trigger Type.......................... Software Time Base .............................. 5 ms/div Trigger Source .............................. Ch1 Trigger Level ...................................... 0 Trigger Slope ............................. Rising

Time interval 𝑑𝑑




From these values, calculate the phase shift between the phasors of the source voltage 𝑈𝑈𝑅𝑅 and resistor voltage 𝑈𝑈𝑅𝑅, using the source voltage phasor as the reference.

Phase shift = °

Phase angle 𝜃𝜃𝑈𝑈𝑅𝑅 = 0°

Phase angle 𝜃𝜃𝑈𝑈𝑅𝑅 = −42.07°

Phase shift = 42.07° lagging or −42.07°

15. Compare the phase shift you determined from the voltage sine waves to the phase shift you measured from the corresponding voltage phasors. Are both values close to each other?

Yes No

Yes

Measuring the phase shift between two voltage sine waves in a resistor-capacitor (RC) circuit

In this section, you will replace the inductor used in the previous section by a capacitor. You will use an oscilloscope to determine the phase shift between the two voltage sine waves. Then, you will use a phasor analyzer to measure the phase shift between the source voltage phasor and the resistor voltage phasor, and compare it to the phase shift you determined from the voltage waveforms.

a As you will see later, due to the presence of a capacitor in the circuit, the circuit current leads the source voltage. As a result, the resistor voltage 𝑈𝑈𝑅𝑅 is out of phase with respect to the source voltage 𝑈𝑈𝑅𝑅.

16. In LVDAC-EMS, turn the ac power source off.

17. Modify the circuit so that it is as shown in Figure 1-16 (replace the inductor by a capacitor). This circuit contains a resistor 𝑅𝑅 and a capacitor 𝐶𝐶. Capacitors are studied in the next unit of this manual.

Figure 1-16. AC circuit with a resistor and a capacitor.

𝑈𝑈𝑅𝑅 100 V

𝐶𝐶

𝑅𝑅 314 Ω U2 U1



The value of capacitor 𝐶𝐶 in the circuit of Figure 1-16 is referred to as the capacitance and is expressed in microfarads (μF). The capacitance value to be used is 5.06 μF.

Make the necessary connections and switch settings on the Resistive Load in order to obtain the resistance values required.

a Appendix C of this manual lists the switch settings to implement on the Resistive Load in order to obtain various resistance values.

Make the necessary connections and switch settings on the Capacitive Load in order to obtain the capacitance value required.

a If necessary, ask your instructor to assist you to obtain the capacitance value required.

18. In LVDAC-EMS, turn the ac power source on. Readjust the ac power source voltage 𝑈𝑈𝑅𝑅 (indicated by meter U1) so that it is equal to 100 V.

19. In LVDAC-EMS, use the oscilloscope to measure the period 𝑇𝑇 of the source voltage 𝑈𝑈𝑅𝑅. Record the value below.

Period 𝛵𝛵 = ms

Period 𝛵𝛵 = 20.04 ms. The results are shown in the following figure.

Period 𝑻𝑻 of the waveform of the source voltage 𝑼𝑼𝑺𝑺 at a frequency of 50 Hz.

20. In LVDAC-EMS, use the oscilloscope to measure the period 𝑇𝑇 of the resistor voltage 𝑈𝑈𝑅𝑅. Record the value below.



Period 𝑇𝑇

Period 𝑇𝑇




Period 𝑻𝑻 of the waveform of the resistor voltage 𝑼𝑼𝑺𝑺 at a frequency of 50 Hz.

21. Compare the period 𝑇𝑇 of the resistor voltage 𝑈𝑈𝑅𝑅 measured in the previous step with the period 𝑇𝑇 of the source voltage 𝑈𝑈𝑅𝑅 recorded in step 19. Are the values close to each other?

Yes No

Yes

22. In LVDAC-EMS, use the oscilloscope to measure the time interval 𝑑𝑑 between the waveforms of the source voltage 𝑈𝑈𝑅𝑅 and resistor voltage 𝑈𝑈𝑅𝑅. Record the value below.

Time interval 𝑑𝑑 = ms


Period 𝑇𝑇

Period 𝑇𝑇



Time interval 𝑑𝑑 = 3.43 ms. The results are shown in the following figure.

Time interval 𝒅𝒅 between the waveforms of the source voltage 𝑼𝑼𝑺𝑺 and resistor voltage 𝑼𝑼𝑹𝑹 at a frequency of 50 Hz.

23. Using Equation (1-11), calculate the phase shift between the source voltage 𝑈𝑈𝑅𝑅 and the resistor voltage 𝑈𝑈𝑅𝑅. Consider the source voltage waveform as the reference.

Phase shift = °

Phase shift = 𝑑𝑑𝛵𝛵∙ 360° = 3.43 ms

20.0 ms∙ 360° = 61.7°

Phase shift = 61.7° leading or 61.7°

24. Is the resistor voltage 𝑈𝑈𝑅𝑅 leading or lagging the source voltage 𝑈𝑈𝑅𝑅?

The resistor voltage 𝑈𝑈𝑅𝑅 is leading the source voltage 𝑈𝑈𝑅𝑅.

25. In LVDAC-EMS, make sure that the source voltage 𝑈𝑈𝑅𝑅 (input U1) is selected as the reference phasor in the phasor analyzer. Measure the phase angles 𝜃𝜃𝑈𝑈𝑅𝑅 and 𝜃𝜃𝑈𝑈𝑅𝑅 of the voltage phasors.






Ex. 1-2 – Phase Angle and Phase Shift Conclusion


From these values, calculate the phase shift between the phasors of the source voltage 𝑈𝑈𝑅𝑅 and resistor voltage 𝑈𝑈𝑅𝑅, using the source voltage phasor as the reference.

Phase shift = °

Phase angle 𝜃𝜃𝑈𝑈𝑅𝑅 = 0°

Phase angle 𝜃𝜃𝑈𝑈𝑅𝑅 = 63.17°

Phase shift = 63.17°

26. Compare the phase shift you determined from the voltage sine waves to the phase shift you measured from the corresponding voltage phasors. Are both values close to each other?

Yes No

Yes

27. In LVDAC-EMS, turn the ac power source off.

28. Close LVDAC-EMS.

29. Turn the 4 Quadrant Power Supply and Dynamometer Controller off.

30. Turn the AC 24 V Power Supply off.

31. Turn electric power off at your workstation, if applicable. Remove all circuit connections, finishing with the equipment earthing connections. Return all equipment to its storage location.

In this exercise, you saw how the phase angle modifies the value of a sine wave at time 𝑡𝑡 = 0, and thus, the position in time of the sine wave. You observed the effects of positive and negative phase angles on the relative position in time of a sine wave. You were introduced to the notion of phase shift. You learned how to calculate and measure the phase shift between two sine waves and to differentiate between a lagging and a leading phase shift.

1. What is the effect of the phase angle on the graphical representation of a sine wave?

The phase angle determines the value of a sine wave when 𝑡𝑡 = 0 s, and thus, the position in time of the sine wave.

CONCLUSION

REVIEW QUESTIONS

Ex. 1-2 – Phase Angle and Phase Shift Review Questions


2. A sine wave has a phase angle 𝜃𝜃 of 72°. Will this sine wave reach its maximum value before, after or at the same time as a second waveform having a phase angle 𝜃𝜃 of -18°?

The sine wave with the phase angle 𝜃𝜃 of 72° will reach its maximum value before the sine wave having a phase angle 𝜃𝜃 of −18°.

3. Given the following two sine wave equations:

𝑈𝑈(𝑡𝑡) = 8 sin 20𝑡𝑡 +78°

𝑈𝑈(𝑡𝑡) = 40 sin 20𝑡𝑡 + 43°

Calculate the phase shift between these two sine waves, considering the first sine wave as the reference. Indicate also whether the second sine wave is lagging or leading the reference sine wave.

Phase shift = 43° − 78° = −35°

The second sine wave is lagging the reference (first) sine wave.

4. When calculating the phase shift between two sine waves, which of the following parameters do the two sine waves need to have in common: phase angle, amplitude, frequency, or period? Why?

To calculate the phase shift between two sine waves, it is necessary for them to have the same frequency and period. Amplitude and phase angle have no effect on the duration of the sine wave cycle.

5. Consider two sine waves with the same frequency. They both have a period 𝛵𝛵 of 50 ms. The second sine wave reaches its maximum positive value 8 ms after the first. Calculate the phase shift between the two sine waves, considering the first one as the reference.

Phase shift = 𝑑𝑑𝛵𝛵∙ 360° = 8 ms

50 ms∙ 360° = 57.6°

Since the second sine wave is lagging the reference, the value of the phase shift is negative, thus:

Phase shift = −57.6° or 57.6° lagging


Bibliography

Boylestad, Robert L., Introductory Circuit Analysis, 11th ed., Upper Saddle River: Prentice Hall, 2006, ISBN 978-0131730441.

Wildi, Theodore, Electrical Machines, Drives, and Power Systems, 6th ed., Upper Saddle River: Prentice Hall, 2005, ISBN 978-0131776913.

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